Properties

Label 1225.2
Modulus $1225$
Conductor $1225$
Order $420$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1225)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([21,260]))
 
pari: [g,chi] = znchar(Mod(2,1225))
 

Basic properties

Modulus: \(1225\)
Conductor: \(1225\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(420\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1225.bv

\(\chi_{1225}(2,\cdot)\) \(\chi_{1225}(23,\cdot)\) \(\chi_{1225}(37,\cdot)\) \(\chi_{1225}(53,\cdot)\) \(\chi_{1225}(58,\cdot)\) \(\chi_{1225}(72,\cdot)\) \(\chi_{1225}(88,\cdot)\) \(\chi_{1225}(102,\cdot)\) \(\chi_{1225}(123,\cdot)\) \(\chi_{1225}(137,\cdot)\) \(\chi_{1225}(142,\cdot)\) \(\chi_{1225}(158,\cdot)\) \(\chi_{1225}(163,\cdot)\) \(\chi_{1225}(172,\cdot)\) \(\chi_{1225}(198,\cdot)\) \(\chi_{1225}(212,\cdot)\) \(\chi_{1225}(228,\cdot)\) \(\chi_{1225}(233,\cdot)\) \(\chi_{1225}(242,\cdot)\) \(\chi_{1225}(247,\cdot)\) \(\chi_{1225}(277,\cdot)\) \(\chi_{1225}(298,\cdot)\) \(\chi_{1225}(303,\cdot)\) \(\chi_{1225}(317,\cdot)\) \(\chi_{1225}(333,\cdot)\) \(\chi_{1225}(338,\cdot)\) \(\chi_{1225}(347,\cdot)\) \(\chi_{1225}(352,\cdot)\) \(\chi_{1225}(387,\cdot)\) \(\chi_{1225}(403,\cdot)\) ...

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\((1177,101)\) → \((e\left(\frac{1}{20}\right),e\left(\frac{13}{21}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\(-1\)\(1\)\(e\left(\frac{61}{420}\right)\)\(e\left(\frac{407}{420}\right)\)\(e\left(\frac{61}{210}\right)\)\(e\left(\frac{4}{35}\right)\)\(e\left(\frac{61}{140}\right)\)\(e\left(\frac{197}{210}\right)\)\(e\left(\frac{59}{105}\right)\)\(e\left(\frac{109}{420}\right)\)\(e\left(\frac{53}{140}\right)\)\(e\left(\frac{61}{105}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{420})$
Fixed field: Number field defined by a degree 420 polynomial